Calculating Darboux Integrals for a Piecewise Function on the Interval [0,b]

[gutnedawg]

Homework Statement

Let f(x) = x for rational x anf f(x) = 0 for irrational x. Calculate the upper and lower Darboux integrals for f on the interval [0,b]. Is f integrable on [0,b]

Homework Equations

The Attempt at a Solution

I'm thinking that f is no integrable but I'm just sketchy with Darboux integrals and need some pointing in the correct direction/explanation

 

[ystael]

To compute the lower and upper Darboux sums for $$f$$ corresponding to any partition of $$[0, b]$$, you will need to know lower and upper estimates for $$f$$ on subintervals of $$[0, b]$$. Therefore, let $$[l, r]$$ be any subinterval of $$[0, b]$$. What is the lower estimate $$\inf \{ f(x) \mid x \in [l, r] \}$$? What is the upper estimate $$\sup \{ f(x) \mid x \in [l, r] \}$$?